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  • Russian children work with Maple - view





    Alsu Gibadullina, math teacher

    Secondary and high school # 57, Kazan, Russia


    In recent years Russia actively promoted and implemented the so-called inclusive education (IE). According to the materials of Alliance of human rights organizations “Save the children”: "Inclusive or included education is the term used to describe the process of teaching children with special needs in General (mass) schools. In the base of inclusive education is the ideology that ensures equal treatment for everybody, but creates special conditions for children with special educational needs. Experience shows that any of the rigid educational system some part of the children is eliminated because the system is not ready to meet the individual needs of these children in education. This ratio is 15 % of the total number of children in schools and so retired children become separated and excluded from the overall system. You need to understand that children do not fail but the system excludes children. Inclusive approaches can support such children in learning and achieving success. Inclusive education seeks to develop a methodology that recognizes that all children have different learning needs tries to develop a more flexible approach to teaching. If teaching and learning will become more effective as a result of the changes that introduces Inclusive Education, all children will win (not only children with special needs)."

    There are many examples of schools that have developed their strategy implementation of IE, published many theoretical and practical benefits of inclusion today. All of them have common, immaterial character. There is no description of specific techniques implementing the principles of IO in the teaching of certain disciplines, particularly mathematics. In this paper we propose a methodology that can be successfully used as in “mathematical education for everyone", also for the development of scientific creativity of children at all age levels of the school in any discipline.

    According to the author, one of the most effective methodological tools for education is a computer mathematics (SCM, SSM). Despite the fact that the SCM were created for solving problems of higher mathematics, their ability can successfully implement them in the school system. This opinion is confirmed by more than 10–year-old author's experience of using the package Maple in teaching mathematics. At first it was just learning the system and primitive using its. Then author’s interactive demonstrations, e-books, programs of analytical testing were created by the tools packages. The experience of using the system Maple in teaching inevitably led to the necessity to teach children to work with it. At first worked a club who has studied  the principles of the package’s work, which eventually turned into a research laboratory for the use of computer technology. Later on its basis there was created the scientific student society (SSS) “GEODROMhic" which operates to this day. The main idea and the ultimate goal of SSS – individual research activities on their interests with the creation of the author electronic scientific journals through the use of computer mathematics Maple. The field of application of the package was very diverse: from mathematics to psychology and cultural phenomena. SSS’s activity is very high: they are constantly and successfully participate in intellectual high-level activities (up to international). Obviously, not every SSS’s member reaches high end result. However, even basic experience in scientific analysis, modeling, intelligent  using of the computer teaches the critical thinking skills, evokes interest to new knowledge, allows you to experience their practical value, gives rise to the development of creative abilities. As a result, the research activity improves intellectual culture, self-esteem and confidence, resistance to external negative influence. It should be noted, however, that members of the scientific societies are not largely the so-called "gifted", than ordinary teenagers with different level of intellectual development and mathematical training. With all this especially valuable is that the student is dealing with mathematical signs and mathematical models, which contributes to the development of mathematical thinking.

    From 2007 to 2012. our school (№. 57 of Kazan) was the experimental platform of the Republican study SKM (Maple) and other application software in the system of school mathematical education under the scientific management of Professor Yu. G. Ignatyev of Kazan state University (KF(P)U).

    Practical adaptation of computer mathematics and other useful information technologies to the educational process of secondary schools passed and continues to work in the following areas:

    1. The creation of a demonstration support of different types of the lessons;
    2. The embedding of computing to the structure of practical trainings;
    3. In the form of additional courses - studying of computer applications through which you can conduct a research of the mathematical model and create animated presentation videos, web-pages, auto-run menu;
    4. Students’ working on individual creative projects:
    • construction of computer mathematical models;
    • creating author's programs with elements of scientific researches;
    • students create interactive computer-based tutorials;
    • creation of an electronic library of creative projects;
    1. The participation of students  in the annual competitions and scientific conferences for students;
    2. The accumulation and dissemination of new methodological experience.

    Traditionally, the training system has the structure: explanation of a new →  the solution of tasks→ check, self-test and control → planning of the new unit  with using analysis. However, the main task types: 1) elementary, 2) basic, 3) combined, 4) integrated, 5) custom. With the increasing the level of training a number of basic tasks are growing and some integrated tasks become a class of basic. Thus, the library for basic operations is generated. The decision of the educational task occurs on the way of mastering the theoretical knowledge of mathematical modeling: 1) analysis of conditions (and construction drawing), 2) the search for methods of solution, 3) computation, 4) the researching.

    To introduce computer mathematics in this training system, you can:

    • At demonstrations. For example, with Maplе facilities you can create a step-by-step interactive and animated images, which are essentially the exact graphic interpretation of mathematical models.
    • If we have centralized collective control.
    • If students have individual self-control.
    • In the analysis of the conditions of the problem, for the construction and visualization of its model, the study of this model.
    • In the computations.
    • In practical training of different forms.
    • In individual projects with elements of research.

    In the learning process with the use of computer mathematics in the school a library of themed demonstrations, tasks of different levels and purpose, programs, analytical testing, research projects is generated. With all this especially valuable is that the student is dealing with mathematical signs and mathematical models. Addiction to them processed in the course of working with them it’s unobtrusive, naturally, organically.

    Mathematical modelling (MM) is increasingly becoming an important component of scientific research. Today's powerful engineering tools allow to carry out numerous computer experiments, deep and full enough of exploring the object, without significant cost painless. Thus provided the advantages of theoretical approach, and experiment. The integration of information technology and      MM method is effective, safe and economical. This explains its wide distribution and makes unavoidable component of scientific and technical progress.

    Modeling is a natural process for people, it is present in any activity. The introduction with nature by man occurs through constant  modeling of situation, comparing with the basic models and past experience by them. Method for modeling, abstraction as a method of understanding the world is therefore  an effective method of learning. Training activities associated with the creative transformation of the subject. The main feature of educational training activities is the systematic solution of the educational problems. The connection of the principles of developmental education, mathematical modeling, neurophysiological mechanisms of the brain and experience with Maple leads to the following conclusions: the method of mathematical modeling is not only scientific research but also the way of development of thinking in general; computer and mathematical environment (Maple), which is a powerful tool for scientific simulation can be considered as the elementary analogue of the brain. These qualities of computer mathematics led to the idea of using it not only as an effective methodological tool but as a means of nurturing the thinking and development of mental functions of the brain. To study this effect the school psychologist conducted a test, which confirm the observations: the dynamics of intellectual options students  who working with Maple compares favorably with peers. In the process of doing computer math, in particular Maple, are involved in complex different mental functions. It is in the inclusion of all mental functions is the essence of integration of learning, its educational character. And this, in turn, contributes to the solution of moral problems.

    Long-term work with computer mathematics led to the idea to use it as a tool for psychological testing. One of the projects focuses on the psychology and contains authors Maple–tests to identify the degree of development of different mental functions. Interactive mathematical environment  gives a wide variability and creative testing capabilities. Moreover, Maple–test can be used not only as diagnostic but also as educational, and corrective. This technology was tested in one of psycho neurological dispensaries a few years ago.

    Currently, one of the author's students, the so-called "homeworkers", the second year is a young man with a diagnosis, categories F20, who does not speak and does not write independently. It was         impossible to get feedback from him and have basic training until then author have begun to apply computer-based tools, including system Maple. Working with the computer tests and mathematical objects helps to see not only the mental and even the simplest thinking movement, but also emotional movement!

    In general, the effectiveness of the implementation in the structure of educational process of secondary school of new organizational forms of the use of computers, based on the application of the symbolic mathematics package Maple, computer modeling and information technology, has many aspects, here are some of them:

    • goals of education and math in particular;
    • additional education;
    • methodical and professional opportunities;
    • theoretical education;
    • modeling;
    • scientific creativity;
    • logical language;
    • spatial thinking, the development of the imagination;
    • programming skills;
    • the specificity of technical translation;
    • differentiation and individualization of educational process;
    • prospective teaching, the continuity of higher and secondary mathematics education;
    • development of creative abilities, research skills;
    • analytical thinking;
    • mathematical thinking;
    • mental diagnosis;
    • mental correction.

           According to the author, the unique experience of the Kazan 57–th school suggests that computer mathematics (Maple) is the most effective also universal tool of new methods of inclusion. In recent decades, there are more children with a specific behavior, with a specific perception, not able to focus, with a poor memory, poor thinking processes. There are children, emotionally and intellectually healthy, or even ahead of their peers in one team together with them. High school should provide all the common core learning standards. It needed a variety of programs and techniques, as well as specialists who use them. Due to its remarkable features, computer mathematics, in particular Maple, can be used or be the basis of the variation of methods of physico-mathematical disciplines of inclusive education.


    Now in English   KozlovaAV.PDF


    In Russian

    Авторский опыт использования математической системы Maple и других компьютерных инструментов в школьном научном обществе

     Арина Козлова

    E-mail:; МБОУ «Школа  № 57» Кировского района г.Казани, 10 класс

     Научный руководитель –

    Гибадуллина Алсу, учитель математики МБОУ «Школа  № 57» Кировского района г.Казани;


     Аннотация. Рассмотрен авторский опыт использования математической системы Maple и других компьютерных инструментов для создания научно-популярных проектов физико-математического направления в рамках школьного научного общества.


    На протяжении более 10 лет наша школа наряду с различными информационными технологиями работает с системой компьютерной математики Maple. Один из аспектов этой деятельности  –  научное общество учащихся «ГЕОДРОМчик», научным руководителем которого является учитель математики Гибадуллина А.И. Направления деятельности ученического научного общества – знакомство с пакетом Maple; освоение компьютерных инструментов, позволяющих работать с графикой, видео, создавать интерактивные меню; работа над индивидуальными научно-популярными проектами и создание авторских тематических электронных журналов, содержащих элементы научного исследования и математического моделирования. Компьютерная математика находит все более широкое применение – от научных исследований до продукции масскультур. Математическое моделирование проникло и в сферу создания рисунка, и в киноиндустрию. Изучение и использование учащимися нашего школьного общества символьных систем, в частности Maple, – это попытка приобщиться к современной мировой культуре компьютерного математического моделирования.

    В данной статье описывается личный опыт автора, как одного из членов школьного НОУ.  

    Знакомство с математической системой Maple началось с работы над проектом «Построение анимированной математической 3D-модели открывающейся книги» в 6-ом классе. Этот проект представляет собой создание пространственного анимированного изображения открывающейся книги средствами аналитической геометрии. В среде Maple была построена поэтапная программа получения этого изображения (таблицы 1 и 2). 

    Таблица 1. Фрагмент программы получения анимированного изображения. 

    > restart:

    Подключение к дополнительным библиотекам

    > with(plots):

    > with(plottools):

    Построение одной из страниц:

    s1:= polygon([[0,0.01,0],[1,0.01,0],[1,1,0],[0,1,0]], thickness=1,color=orange):

    Визуализация совокупных элементов книги:

    display(k11,s1,s2,s3,s4,s5,s6,s7,s8,s9,s10,k0,k27, title="KNIGA",scaling=constrained);

    Поворот и анимация открывания обложки:



    > anm:=seq(rotate(k11,t*Pi/9,[[0,0,0.29],[1,0,0.29]]),t=0..10):

    > anim:=plots[display](anm,insequence=true):

    > plots[display](kn_1,anim,scaling=constrained);









     Рис. 1. Кадры анимации книги

    Следующий проект, выполненный в среде Maple совместно с Нигометзяновой Эльзой в 7-ом классе, – короткометражный мультфильм «Колобок в лесу».

    a)      b)   

     Рис. 2. Кадры анимации мультфильма

    В 8-ом классе велась работа по техническому переводу сайта компании Waterloo Maple Inc. [3]. Как известно, такой перевод имеет свои особенности, которые не предусмотрены в школьной программе по изучению английского языка, поэтому опыт такой работы способствует совершенствованию владения английским языком.

    В 9-ом классе началась работа над электронным журналом по космологии «Вселенная: теория и факты». Черные дыры Вселенной – один из самых загадочных и любопытных для человека объектов. Их изучение привело к интересу к астрофизике вообще. Знакомство с понятием черной дыры неизбежно вынудило изучать строение Вселенной и ее геометрии [9, 10, 11, 12]. Пришлось осмысливать сложнейшие фундаментальные понятия, теории, а также элементы высшей математики [1, 5, 6, 7, 8]. Чтобы хотя бы попытаться понять огромный объем, казалось бы, беспорядочной информации, нужно было ее анализировать и систематизировать. И тогда возникла идея проекта – авторского электронного журнала. Тем более складывается парадоксальная ситуация: астрофизика бурно развивается, проникая практически во все сферы нашей жизни, а предмета астрономии в школе нет. Поэтому такой проект мог бы восполнить этот досадный пробел и помочь школьникам – и не только – в познании Вселенной. Журнал имеет следующие разделы: Вселенная, черные дыры, белые дыры, глоссарий, теории, неевклидовы геометрии, видео-опыты, интересные факты, ссылки, использованные ресурсы. Один из разделов журнала составляют Maple-разработки, в частности, визуализированная модель искривления пространства.

    Далее приводится Maple-программа (табл. 2) построения визуализации деформации плоскости под шаром определенного размера. Используются библиотеки <plots> и <plottools> пакета.

     Таблица 2.  Maple–программа визуализации деформации плоскости. 


    Команда и результат

    Функция глубины "ямы"

    ( a - ширина "ямы", b - глубина )



    Вводим параметры:

    h - влияет на размеры тела-шарика и связывает их с шириной "ямы" ;

     k - влияет на диапазон площади вокруг "ямы"

    h:=1:  k:=1:


    Задание параметрическое прямой на поверхности (плоскости)

    L0:=(m,n)->plot3d([0,r,f(r,m,n)], phi = -2*Pi ..2*Pi, r = -10k*h..10+k*h, scaling=CONSTRAINED,

    numpoints=10000, color=blue,thickness=4):

    Задание параметрическое поверхности (плоскости) путем кручения прямой

    P0:=(m,n)->plot3d([r*cos(phi),r*sin(phi),f(r,m,n)], phi= 0..2*Pi,r=-8k*h..8+k*h, scaling=CONSTRAINED, numpoints=3000, style=POINT, color=blue):

    Задание анимации искривления прямой

    L:=plots[display](seq(L0(h,i),i=0..10+k*h), insequence=true):                    l:=plots[display](L,insequence=true):

    Задание анимации искривления плоскости

    p:=plots[display](seq(P0(h,i),i=0..10+k*h), insequence=true):


    Задание анимация шарика ( тела, обладающего массой )


    with(plottools):sp:=seq(sphere([0,0,-i-1.5*f(h,h,h)], f(h,h,h), style=HIDDEN,color=red),i=0..10+k*h):     



    Совмещение всех компонентов модели визуализации


     При h:=1:  k:=1:

     1)      2)      3)   

    4)      5)      6)   


    При h:=5:  k:=1:

    7)      8)      9)   

    Рис. 3. Кадры анимации при заданных параметрах.

    Долго подбиралась функция глубины "ямы". Наконец, была найдена – это стало понятно после просмотра лекции А.Линде, где говорится об экспоненциальных процессах [13].

    Меняя только параметры h и k (задающие размеры шара и ширины «ямы») и прокручивая программу снова, меняется и визуализация. Надо заметить, что построена всего лишь математическая модель визуализации, а не самого процесса.

    Этот раздел предполагается пополнять новыми разработками, выполненными в среде Maple.

    Журнал имеет удобную систему ссылок и организован так, что его можно оперативно обновлять. Астрофизика бурно развивается, поэтому журнал не потеряет своей актуальности.


    В течение 4-х лет занятий в научном обществе авторские проекты были представлены на различных сайтах, конкурсах, конференциях, форумах федерального и международного уровней:

    • сайт е в разделе студенческих работ [4];
    • Конкурс исследовательских и творческих работ «Нобелевские надежды КНИТУ»
    • Республиканский конкурс «Арт-дебют»
    • V Международная ассамблея школьников (участие и публикация) [2]
    • Всероссийский Горчаковский форум в г.Санкт-Петербург
    • Поволжская научной конференция учащихся им. Н.И.Лобачевского
    • Всероссийский фестиваль «Нескучная наука» в г.Санкт-Петербург
    • Пост н.р. Гибадуллиной А.И. на сайте компании Maplesoft

      Использованная литература

     [1] Матросов А.В. Maple 6: Решение задач высшей математики и механики: Практическое руководство. – СПб.: БХВ – Петербург, 2001 г. – 528 с.

    [2] V Международная Интеллектуальная Ассамблея школьников: сборник научно-исследовательских работ / Отв. ред. М. В. Волкова – Чебоксары: НИИ педагогики и психологии, 2012 – 136с. (с. 44–45)

    [3] Сайт компании Maplesoft. – Режим доступа:

    [4] Сайт <> / Архив студенческих работ – Режим доступа:

    [5] Высшая математика: Учеб. Пособие для студентов пед. ин-тов по спец. 2120 «Общетехн. дисциплины и труд» / Г. Луканкин, Н. Мартынов, Г. Шадрин, Г. Яковлев; Под. ред. Г.Н. Яковлева. – М.: Просвещение, 1988. – 431 с.: ил.

    [6] Справочник по высшей математике / М. Я. Выгодский. – М.: ООО «Издательство Астрель»: ООО «Издательство АСТ», 2002. – 992 с.: ил.

    [7] Математический словарь высшей школы: Общ. часть/В. Т. Воднев, А. Ф. Наумович, Н.Ф. Наумович; Под ред. Ю.С. Богданова. – 2-е изд. – М.:Изд-во МПИ, 1988 – 527 с., ил.

    [8] Толковый математический словарь. Основные термины: около 2500 терминов. – М.: Рус. яз., 1989. – 244 с., 186 ил.

    [9] Открываем неевклидову геометрию. Кн. для внеклас. чтения учащихся 9-10 кл. сред. шк. – М.: Просвещение, 1988. – 126 с.: ил. – (Мир Знаний).

    [10] Геометрия: Учебник для вузов. – СПб.: Издательство «Лань», 2003. – 416 с., ил. – (Учебники для вузов. Специальная литература)

    [11] Основания геометрии: Учебн. пособие для вузов. – М.: Наука. Гл. ред. физ.-мат. лит., 1987. – 288 с.

    [12] Обзорные лекции по геометрии к государственному экзамену по математике, Х семестр, курс лекций с примерами решений задач (в помощь выпускнику), проф. Ю.Г. Игнатьева. Программный продукт BIBLIO профессора Ю.Г. Игнатьева, Казань 2002 г.

    [13] Видеозапись лекции Андрея Дмитриевича Линде, Стэнфордский университет (США), профессор «Многоликая Вселенная», прямая ссылка:

    Ibragimova Evelina, 6 class,
    school № 57, Kazan

    The manual with examples
    ( templates for the solution of )

    The solution of problems on simple interest


    > restart:
    > with(finance);

    [amortization, annuity, blackscholes, cashflows, effectiverate,

    futurevalue, growingannuity, growingperpetuity, levelcoupon,

    perpetuity, presentvalue, yieldtomaturity]

    Team futurevalue (the first installment, rate, period) - the total calculation for a given down payment, interest rate, payments and number of periods.

    Example 1. To the Bank account, the income of which is 15% per annum, has made 24 thousand rubles. How many thousands of rubles will be in this account after a year if no transactions on the account will not be carried out? (The answer: 27.60 thousand rubles.)

    > futurevalue(260,0.40,1);


    > evalf(1000/216);

    > 364*3;


    > u:=fsolve(presentvalue(1e6,x,1250)=950,x)*950;

    u := 5.303626495


    Team presentvalue (future amount, rate, period) - the calculation of the initial input to obtain a specified final amount at an interest rate of charges and the number of periods.

    Example 2. How much you need to put money in the Bank today, so that when the rate of 27% per annum have in the account after 10 years 100000 thousand rubles? (The answer: 9161.419934 rubles.)

    > presentvalue(680,-0.20,1);



    The solution of problems in compound interest

    The solution of problems 
    Using commands <futurevalue> и <presentvalue >
    > restart;
    > with(finance):
    Direct task
    > futurevalue(,0.,);
    `,` unexpected
    The inverse problem
    > presentvalue(,0.,);
    `,` unexpected

    I. Case with the same interest rate every period

    Using the universal formula F = P*(1+r)^n; , where:
    F - the future value (final amount).
    P - the initial payment (current amount).
    r - the interest rate period.
    n - the number of periods.
    This formula for the case with the same interest rate every period

    > restart:
    The task of the formula
    > y:=F=P*((1+r)^n):
    > y;

    F = P (1 + r)

    The job parameters are known quantities
    The interest rate

    > r:=;
    `;` unexpected
    The number of years (periods)
    > n:=3;

    n := 3

    The initial payment (present value)
    > P:=;
    `;` unexpected
    The final amount
    > F:=2.16;

    F := 2.16

    The solution of the equation - the calculation of unknown values (in decimal form)
    > `Unknown`;fsolve(y);




    II. The case of different interest rates for each period

    Formula An = A*(1+1/100*p1)*(1+1/100*p2)*(1+1/100*p3); ... %?(1+1/100*pn); , where
    An - the final amount
    A - the initial payment (current amount at the moment)
    p1, p2, p3, .... pn - interest rate periods
    n - the number of periods

    > restart:
    The task of the formula (need to be adjusted based on the number of periods)
    > y:=An=A*(1+1/100*p1)*(1+1/100*p2)*(1+1/100*p3):
    > y;

    An = A (1 + 1/100 p1) (1 + 1/100 p2) (1 + 1/100 p3)

    The task of the parameters of the known values
    The initial payment (present value)
    > A:=;
    `;` unexpected
    Interest rate periods

    p1 := .30

    p2 := .10

    p3 := .15

    The final amount
    > An:=;
    `;` unexpected
    The solution of the equation - the calculation of unknown values (in decimal form)
    > `Unknown`;fsolve(y);





    Ibragimova Evelina, 6th form,
    school № 57, Kazan






     Ibragimova Evelina, the 6th form

     school # 57, Kazan, Russia

    The Units Converter

    `Conversion formula`;

                        Conversion formula
                        l = n m

    m - shows how many minor units in one major one (coefficient)

    n - the number of major units
                       n := 7/3

    l - the number of minor units

    The result (the desired value)

    That is : there are 2333.3 (or 7000/3 ) minor units in 7/3 major units .


    Other example

    m - shows how many minor units in one major one (coefficient) 

    n - the number of major units 
    l - the number of minor units


    The result (the desired value)

    That is : there are 0.02 (or 1/42) major units in 100 minor units .


    Another example

    m - shows how many minor units in one major one (coefficient) 

    n - the number of major units 

                        n := 2
    l - the number of minor units


    The result (the desired value)

    That is : Coefficient is 100 .

      The geometry of the triangle
      Romanova Elena,  8 class,  school 57, Kazan, Russia

           Construction of triangle and calculation its angles

           Construction of  bisectors
           Construction of medians
           Construction of altitudes

    > restart:with(geometry):      

    The setting of the height of the triandle and let's call it "Т"
    > triangle(T,[point(A,4,6),point(B,-3,-5),point(C,-4,8)]);


            Construction of the triangle
    > draw(T,axes=normal,view=[-8..8,-8..8]);

    Construction of the triangle АВС

    > draw({T(color=gold,thickness=3)},printtext=true,axes=NONE);     
    Calculation of the distance between heights А and В - the length of a side АВ

    > d1:=distance(A,B);

                               d1 := sqrt(170)

            Calculation of the distance between heights В and С - the length of a side ВС
    > d2:=distance(B,C);

                               d2 := sqrt(170)

           The setting of line which passes through two points А and В
    > line(l1,[A,B]);


           Display the equation of line l1
    > Equation(l1);
    > x;
    > y;

                             -2 + 11 x - 7 y = 0

            The setting of line which passes through two points А and С
    > line(l2,[A,C]);


           Display the equation of line l2
    > Equation(l2);
    > x;
    > y;

                              56 - 2 x - 8 y = 0

             The setting of line which passes through two points В and С
    > line(l3,[B,C]);


            Display the equation of line l3
    > Equation(l3);
    > x;
    > y;

                              -44 - 13 x - y = 0

            Check the point А lies on line l1
    > IsOnLine(A,l1);


            Check the point А lies on line l1
    > IsOnLine(B,l1);


            Calculation of the andle between lines l1 and l2
    > FindAngle(l1,l2);


            The conversion of result to degrees
    > b1:=convert(arctan(97/14),degrees);

                                   arctan(--) degrees
                         b1 := 180 ------------------

            Calculation of decimal value of this angle
    > b2:=evalf(b1);

                          b2 := 81.78721981 degrees

            Calculation of the andle between lines l1 and l3
    > FindAngle(l1,l3);


           The conversion of result to degrees
    > b3:=convert(arctan(97/99),degrees);

                                   arctan(--) degrees
                         b3 := 180 ------------------

            Calculation of decimal value of this angle
    > b4:=evalf(b3);

                          b4 := 44.41536947 degrees

           Calculation of the angle between lines l2 and l3
    > FindAngle(l2,l3);


           The conversion of  result to degrees
    > b5:=convert(arctan(97/71),degrees);

                                   arctan(--) degrees
                         b5 := 180 ------------------

            Calculation of decimal value of  this angle
    > b6:=evalf(b5);

                          b6 := 53.79741070 degrees

            Check the sum of all the angles of the triangle
    > b2+b4+b6;

                             180.0000000 degrees

            Analytical information about the point А
    > detail(A);
       name of the object: A
       form of the object: point2d
       coordinates of the point: [4, 6]
              Analytical information about the point В
    > detail(B);
       name of the object: B
       form of the object: point2d
       coordinates of the point: [-3, -5]
              Analytical information about the point С
    > detail(C);
       name of the object: C
       form of the object: point2d
       coordinates of the point: [-4, 8]

       The setting of heights of the triangle points A,B,C and let's call it "Т"

    > triangle(ABC, [point(A,7,8), point(B,6,-7), point(C,-6,7)]):
            The setting of the bisector of angle А in triandle АВС
    > bisector(bA, A, ABC);


            Analytical information about the bisector of angle А in the triandle
    > detail(bA);
       name of the object: bA
       form of the object: line2d
       assume that the name of the horizonal and vertical                    axis are _x and _y
       equation of the line: (15*170^(1/2)+226^(1/2))*_x+(-13*226^(1/2)-170^(1/2))*_y+97*226^(1/2)-97*170^(1/2) = 0

            Construction of the triangle
    > draw(ABC,axes=normal,view=[-8..8,-8..8]);

     Construction of the triangle ABC

    > draw({ABC(color=gold,thickness=3)},printtext=true,axes=NONE);     

     Construction of the bisector of angle А

    > draw({ABC(color=gold,thickness=3),bA(color=green,thickness=3)},printtext=true,axes=NONE);    

    The setting of the bisector of angle В in the triangle АВС

    > bisector(bB, B, ABC);


           Analytical information about the bisector of angle B in the triandle
    > detail(bB);
       name of the object: bB
       form of the object: line2d
       assume that the name of the horizonal and vertical                    axis are _x and _y
       equation of the line: (-15*340^(1/2)-14*226^(1/2))*_x+(-12*226^(1/2)+340^(1/2))*_y+97*340^(1/2) = 0

             Construction of the bisector of angle В

        The setting of the bisector of angle С in the triangle АВС

    > bisector(bC, C, ABC);


            Analytical information about the bisector of angle С in the triangle
    > detail(bC);
       name of the object: bC
       form of the object: line2d
       assume that the name of the horizonal and vertical                    axis are _x and _y
       equation of the line: (14*170^(1/2)-340^(1/2))*_x+(13*340^(1/2)+12*170^(1/2))*_y-97*340^(1/2) = 0

            Construction of the bisector of angle С

     Calculation of the point of intersection of the bisectors and let's call it "О"

    > intersection(O,bA,bB,bC);coordinates(O);


         7 sqrt(85) - 3 sqrt(2) sqrt(113) + 3 sqrt(85) sqrt(2)
      [2 -----------------------------------------------------,
           sqrt(85) sqrt(2) + sqrt(2) sqrt(113) + 2 sqrt(85)

              -16 sqrt(85) - 7 sqrt(2) sqrt(113) + 7 sqrt(85) sqrt(2)
            - -------------------------------------------------------]
                 sqrt(85) sqrt(2) + sqrt(2) sqrt(113) + 2 sqrt(85)

           Construction of the bisectors and  marking of the point of intersection  "О" in the triandle
    > restart:
    > with(geometry):
           The setting of the heights of the triangle points A,B,C and let's call it "Т"
    > point(A,7,8),point(B,6,-7),point(C,-6,7);

                                   A, B, C

            Let's call "Т1"
    > triangle(T1,[A,B,C]);


            Construction of "Т1"
    > draw(T1(color=gold,thickness=3),axes=NONE,printtext=true);
      The setting of the median from the point В in the trianglemedian(mB,B,T1,B1);
    > median(mb,B,T1);



            Construction of the median from the point В
    > draw({T1(color=gold,thickness=3),mB(color=green,thickness=3),mb},printtext=true,axes=NONE);

    The setting of the median from the point А in the trianglemedian(mA,A,T1,A1);
    > median(ma,A,T1);



            Construction of the median from the point А
    The setting of the median from the point С in the trianglemedian(mC,C,T1,C1);
    > median(mc,C,T1);



            Costruction of the median from the point С

    Calculation of the point of  intersection of the median and let's call it "О"



                                  [7/3, 8/3]

            Construction of medians and marking of the point of  intersection "О" in the triangle
    > restart:with(geometry):
    > _EnvHorizontalName:=x:_EnvVerticalName=y:       The setting of the heights of the triangle points A, B, C  and let's call it "Т"
    > triangle(T,[point(A,7,8),point(B,6,-7),point(C,-6,7)]);


           Construction of the triangle
    > draw(T,axes=normal,view=[-8..8,-8..8]);

    The setting of the altitude in the triangle from the point Сaltitude(hC1,C,T,C1);
    > altitude(hC,C,T);



            Analytical information about the altitude hC from the point С in the triangle
    > detail(hC);
       name of the object: hC
       form of the object: line2d
       assume that the name of the horizonal and vertical                    axis are _x and _y
       equation of the line: -99+_x+15*_y = 0

            Construction of the altitude from the point С
    > draw({T(color=gold,thickness=3),hC1(color=green,thickness=3),hC},printtext=true,axes=NONE);     

      The setting of the altitude in the triangle from the point Аaltitude(hA1,A,T,A1);
    > altitude(hA,A,T);



            Analytical information about the altitude hA from the point А in the triangle
    > detail(hA);
       name of the object: hA
       form of the object: line2d
       assume that the name of the horizonal and vertical                    axis are _x and _y
       equation of the line: -28-12*_x+14*_y = 0

            Construction of the altitude from the point А
    >draw({T(color=gold,thickness=3),hC1(color=green,thickness=3),hA1(color=red,thickness=3),hA1},printtext=true,axes=NONE);       The setting of the altitude from the point В

    > altitude(hB1,B,T,B1);
    > altitude(hB,B,T);



            Analytical information about the altitude hB from the point В in the triangle
    > detail(hB);
       name of the object: hB
       form of the object: line2d
       assume that the name of the horizonal and vertical                    axis are _x and _y
       equation of the line: -71+13*_x+_y = 0

            Consruction of the altitude from the point В
     Calculation of the point of intersection of altitudes and let's call it "О"



                                   483  608
                                  [---, ---]
                                   97   97

            Construction of altitudes and marking of the point of intersection "О" in the triangle














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    Maplesoft Product Manager, Online Education Products

      Elena, Liya

      "Researching turkish song: the selection of the main element and its graphic transformations",

       Russia, Kazan, school #57

    The setting and visualization of the melodic line of the song
    > restart:
    > with(plots):with(plottools):
    > p0:=plot([[0.5,9],[1,7],[2,9],[4,11],[6,9],[7,11],[8,7],[10,9],[12,9],[14,9],[16,7],[16.5,9],[17,7],[18,9]],color=magenta):p1:=plot([[18,9],[20,11],[22,9],[23,11],[24,9],[26,11],[28,11],[29.5,8],[30,11],[32,9],[33.5,8],[34,9],[36,7],[37.5,5],[38,9],[40,7],[42,5],[44,5],[46,4],[47,5],[48,2],[50,4],[51,5],[51.5,4],[52,2],[54,4],[56,4],[56.5,5],[57,4],[58,5],[60,7],[62,5],[64,7],[66,5]],color=cyan):
    > p2:=plot([[66,5],[68,5],[69,5],[70,4],[71,5],[71.5,4],[72,2],[73,4],[74,5],[75,7],[76,5],[78,4],[78.5,7],[80,5],[82.5,4],[83.5,4],[84,2],[86,4],[88,4],[90.5,4],[91.5,4]],color=red):
    > p3:=plot([[91.5,4],[92,2],[94,4],[96,4],[96.5,9],[97,7],[98,9],[100,11],[100.5,9],[101,11],[102,9],[104,11],[106,9],[108,9],[109,9],[109.5,9],[110,7],[111,9],[112,7],[113,7],[114,9],[116,11],[116.5,9],[117,11],[118,9],[119.5,11],[120,9],[122.5,9],[124,9],[124.5,9],[125,11],[125.5,9],[126,11],[128,9],[129,7],[130,9],[132,11],[132.5,9],[133,11],[134,9],[136,11],[136.5,9],[138.5,9],[140,9],[140.5,9],[141,11],[141.5,9],[142,11],[143,7],[143.5,7],[144,9],[144.5,9],[145,7],[146,9],[148,11],[148.5,9],[149,11],[150,9],[151.5,11],[152,9],[154.5,9],[156,9],[156.5,9],[157,11],[157.5,9],[158,11],[160,9],[161,7],[162,9],[164,11],[164.5,9],[165,11],[166,9],[168,11],[168.5,9],[171.5,9],[172,9],[172.5,9],[173.5,11],[174,9],[174.5,11],[175,7],[175.5,7],[176,9],[176.5,9],[177,7],[178,9],[180,11],[180.5,9],[181,11],[182,9],[183.5,11],[184,9],[186.5,9],[188,9],[188.5,9],[189,11],[189.5,9],[190,11],[192,9],[192.5,9],[193,7],[194,9],[196,11],[196.5,9],[197,11],[198,9],[200,11],[201.5,9],[202,11],[203,9],[203.5,8],[204,9],[205,7],[205.5,9],[206,11],[207,9],[208,7],[209,8],[209.5,7],[210,9],[211,7],[212,5],[213,5],[213.5,5],[214,9],[215,7],[216,5],[217,5],[217.5,5],[218,7],[219,5],[220,4],[221,4],[221.5,4],[222,7],[223,5],[224,4],[225,4],[227,4],[227.5,4],[228,2],[230,4]],color=blue):
    > p4:=plot([[230,4],[232,4],[232.5,5],[233,4],[234,5],[236,7],[236.5,5],[237,5],[238,9],[240,7],[242.5,5],[244,5],[245,5],[246,4],[246.5,5],[247,4],[248,2],[250,4],[250.5,7],[251,5],[252,4],[254,4],[254.5,7],[255,5],[256,4],[258,4]],color=brown):
    > p5:=plot([[258,4],[259,4],[260,2]],color=green):
    > plots[display](p0,p1,p2,p3,p4,p5,thickness=2);



    The selection of the main melodic element in graph of whole song. The whole song is divided into separate elements - results of transformationss0:=plot([[7,11],[8,7],[10,9],[12,9],[14,9],[16,7],[16.5,9]],color=blue):
    > s1:=plot([[118,9],[119.5,11],[120,9],[122.5,9],[124,9],[124.5,9],[125,11],[125.5,9]],color=blue):
    > s2:=plot([[134,9],[136,11],[136.5,9],[138.5,9],[140,9],[140.5,9],[141,11],[141.5,9]],color=blue):
    > s3:=plot([[150,9],[151.5,11],[152,9],[154.5,9],[156,9],[156.5,9],[157,11],[157.5,9]],color=blue):
    > s4:=plot([[166,9],[168,11],[168.5,9],[171.5,9],[172,9],[172.5,9],[173.5,11],[174,9]],color=blue):
    > s5:=plot([[182,9],[183.5,11],[184,9],[186.5,9],[188,9],[188.5,9],[189,11],[189.5,9]],color=blue):
    > s6:=plot([[250,4],[250.5,7],[251,5],[252,4],[254,4],[254.5,7],[255,5],[256,4]],color=blue):
    > plots[display](s0,s1,s2,s3,s4,s5,s6);
    > s:=plots[display](s0,s1,s2,s3,s4,s5,s6):


    Animated display of grafical transformation of the basic element (to click on the picture - on the panel of instruments appears player - to play may step by step).m0:=plot([[7,11],[8,7],[10,9],[12,9],[14,9],[16,7],[16.5,9]],color=blue):
    > pm:=plot([[118,9],[119.5,11],[120,9],[122.5,9],[124,9],[124.5,9],[125,11],[125.5,9]],color=red,style=line,thickness=4):
    > iop:=plots[display](m0,pm,insequence=true):
    > plots[display](iop,s0);

    > m0_t:=translate(m0,110,0):
    > m0_r:=reflect(m0_t,[[0,9],[24,9]]):
    > plots[display](m0,m0_r,insequence=true);
    > m0r:=plots[display](m0,m0_r,insequence=true):

    > pm0:=plots[display](pm,m0):
    > plots[display](pm0,m0r);

    > m0:=plot([[7,11],[8,7],[10,9],[12,9],[14,9],[16,7],[16.5,9]],color=blue):
    > pn:=plot([[134,9],[136,11],[136.5,9],[138.5,9],[140,9],[140.5,9],[141,11],[141.5,9]],color=blue,thickness=3):
    > iop:=plots[display](m0,pn,insequence=true):
    > plots[display](iop,s0);

    > m0_t1:=translate(m0,126,0):
    > m0_r1:=reflect(m0_t1,[[0,9],[24,9]]):
    > plots[display](m0,m0_r1,insequence=true);
    > m0r1:=plots[display](m0,m0_r1,insequence=true):

    > pm01:=plots[display](pn,m0):
    > plots[display](pm01,m0r1);


    > pm2:=plots[display](pn,pm,m0):
    > plots[display](pm0,m0r,pm01,m0r1);

    > pt_i_1:=seq(translate(pm,5*11*i,0),i=0..4):
    > plots[display](pt_i_1);

    > pm_i:=seq(translate(pm,5*11*i,0),i=0..4):
    > plots[display](pm_i);
    > iop1:=plots[display](pm_i,insequence=true):
    > plots[display](iop1,s0);


    > pm_i_0:=seq(translate(m0_r,5*11*i,0),i=0..4):
    > plots[display](pm_i_0);
    > iop2:=plots[display](pm_i_0,insequence=true):
    > plots[display](iop2,s0);







    Construction of arabesques of melodic line BACH

    Elena, Liya "Construction of arabesques of melodic line BACH", Kazan, Russia, school#57
    > restart:
    > with(plots):with(plottools):

          The setting and visualization of line BACH: B - note b-flat, A - note la, C - note do, H - note si.
    > p0:=plot([[0,1],[2,0],[4,1.5],[6,1]],thickness=4,color=cyan,scaling=constrained);
    >   p0 := PLOT(
    >         CURVES([[0, 1.], [2., 0], [4., 1.500000000000000], [6., 1.]])
    >         COLOUR(RGB, 0, 1.00000000, 1.00000000),
    >         VIEW(DEFAULT, DEFAULT))
    > plots[display](p0);
    > r_i:=seq(rotate(p0,i*Pi/4),i=1..8):
    > p1:=display(r_i,p0):plots[display](p1,scaling=constrained);

    > c1:=circle([0,0],6,color=blue,thickness=2):
    > plots[display](c1,p1,scaling=constrained);
    > p_c:=plots[display](c1,p1,scaling=constrained):

    > pt_i_2:=seq(translate(p1,0,2*6*i),i=0..4):
    > plots[display](pt_i_2,scaling=constrained);
    > pt_i_22:=seq(translate(p1,0,6*i),i=0..4):
    > plots[display](pt_i_22,scaling=constrained);
    > pt_i_222:=seq(translate(p1,0,1/2*6*i),i=0..4):
    > plots[display](pt_i_222,scaling=constrained);

    > pr:=rotate(p1,Pi/8):
    > plots[display](pr,scaling=constrained);
    > plots[display](p1,pr,scaling=constrained);
    > pr_i:=seq(rotate(p1,Pi/16*i),i=0..8):
    > plots[display](pr_i,scaling=constrained);

    > pt_1:=translate(p1,0,2*6):
    > pr_1_i:=seq(rotate(pt_1,Pi/3.5*i),i=0..6):
    > plots[display](pr_1_i,scaling=constrained);
    > pr_11_i:=seq(rotate(pt_1,Pi/5*i),i=0..10):
    > plots[display](pr_11_i,scaling=constrained);
    > pr_111_i:=seq(rotate(pt_1,Pi/6.5*i),i=0..12):
    > plots[display](pr_111_i,scaling=constrained);

    Elena, Liya "Designing of islamic arabesques", Kazan, Russia, school #57

    > restart:
          At the theorem of cosines  c^2 = a^2+b^2-2*a*b*cos(phi);
          In our case  c=a0 ,  a=1 ,  a=b , phi; - acute angle of a rhombus (the tip of the kalam).
          s0 calculated at theorem of  Pythagoras.
         (а0 - horizontal diagonal of a  rhombus, s0 - vertical diagonal of a  rhombus)
    > a:=1:phi:=Pi/4:
    > a0:=sqrt(a^2+a^2-2*a^2*cos(phi));

                           a0 := sqrt(2 - sqrt(2))

    > solve((s0^2)/4=a^2-(a0^2)/4,s0);

                    sqrt(2 + sqrt(2)), -sqrt(2 + sqrt(2))

          The setting of initial parameters : the size of the tip of the pen-kalam and  depending on its - the main module size - point
           (а0 - horizontal diagonal of a  rhombus, s0 - vertical diagonal of a  rhombus)
    > a0:=sqrt(2-sqrt(2)):
    > s0:=sqrt(2+sqrt(2)):
          Connection the graphical libraries Maple
    > with(plots):with(plottools):
          Construction of unit of measure (point) - rhombus - the tip of the kalam
    > p0:=plot([[0,0],[a0/2,s0/2],[0,s0],[-a0/2,s0/2],[0,0]],scaling=constrained,color=gold,thickness=3):
    > plots[display](p0);

    The setting and construction of altitude of alif - the basis of the rules compilation of the proportions      Example, on style naskh altitude of alif amount five points
    > p_i:=seq(plot([[0,0+s0*i],[a0/2,s0/2+s0*i],[0,s0+s0*i],[-a0/2,s0/2+s0*i],[0,0+s0*i]],scaling=constrained,color=black),i=0..4):
    > pi:=display(p_i):
    > plots[display](p_i);
    The setting of appropriate circle of diameter, amount altitude of alifd0:=s0+s0*i:
    > i:=4:
    > d0:=d0:
    > c0:=circle([0,d0/2],d0/2,color=blue):
    > plots[display](p_i,c0);

    Construction of flower by turning "point"r_i:=seq(rotate(p0,i*Pi/4),i=1..8):
    > p1:=display(r_i,p0):plots[display](p1,scaling=constrained);

     The setting of circumscribed circlec1:=circle([0,0],s0,color=blue,thickness=2):
          Construction and the setting of flower inscribed in a circle
    > plots[display](c1,p1,scaling=constrained);
    > p_c:=plots[display](c1,p1,scaling=constrained):

    The setting and construction of arabesque by horizontal parallel transport original flower with different stepspt_i_1:=seq(translate(p1,5*a0*i,0),i=0..4):
    > plots[display](pt_i_1);
    > pt_i_11:=seq(translate(p1,2*a0*i,0),i=0..4):
    > plots[display](pt_i_11);
    > pt_i_111:=seq(translate(p1,a0*7*i,0),i=0..4):
    > plots[display](pt_i_111);

     The setting and construction of arabesque by vertical parallel transport original flower with different stepspt_i_2:=seq(translate(p1,0,2*s0*i),i=0..4):
    > plots[display](pt_i_2);
    > pt_i_22:=seq(translate(p1,0,s0*i),i=0..4):
    > plots[display](pt_i_22);
    > pt_i_222:=seq(translate(p1,0,1/2*s0*i),i=0..4):
    > plots[display](pt_i_222);
     Getting arabesques by turning original flower on different anglespr:=rotate(p1,Pi/8):
    > plots[display](pr);
    > plots[display](p1,pr);

    > pr_i:=seq(rotate(p1,Pi/16*i),i=0..8):
    > plots[display](pr_i);

    > pt_1:=translate(p1,0,2*s0):
    > pr_1_i:=seq(rotate(pt_1,Pi/3.5*i),i=0..6):
    > plots[display](pr_1_i);
    > pr_11_i:=seq(rotate(pt_1,Pi/5*i),i=0..10):
    > plots[display](pr_11_i);
    > pr_111_i:=seq(rotate(pt_1,Pi/6.5*i),i=0..12):
    > plots[display](pr_111_i);